C-0-pi-dx-2-cos-2x-

Question Number 165955 by cortano1 last updated on 10/Feb/22

C = \int_{0}^{π} \frac{dx}{2 + \cos 2 x} = ?

Answered by MJS_new last updated on 10/Feb/22

\underset{0}{\int^{π}} \frac{d x}{2 + \cos 2 x} = 2 \underset{0}{\int^{π / 2}} \frac{d x}{2 + \cos 2 x}

\int \frac{d x}{2 + \cos 2 x} =

[t = \tan x \to d x = \frac{d t}{t^{2} + 1}]

= \int \frac{d t}{t^{2} + 3} = \frac{\sqrt{3}}{3} \arctan \frac{\sqrt{3} t}{3} = \frac{\sqrt{3}}{3} \arctan \frac{\sqrt{3} \tan x}{3} + C

\Rightarrow

\underset{0}{\int^{π}} \frac{d x}{2 + \cos 2 x} = \frac{\sqrt{3} π}{3}

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